Theorem 35.1:

Suppose $U$ and $V$ are subsets of a topological space such that Int$\; U\cup$   Int$\; V = X$. Then there is a long exact sequence

$$\begin{CD} @> >> H_n(U\cap V) @> (\kappa^{\prime}, -\kappa^{\pri... ... H_n(V) @> q_n >> H_n(U\cup V) @>{\delta_n} >> H_{n-1}(U\cap V) @> >> \end{CD}$$